Question

There are many varieties of mushrooms available in the world. One such mushroom ‘Amanita muscaria’ has a upper part which is like red cap (hemispherical) and lower part is like white stem (cylinderical). The hemispherical cap’s radius = 3 cm and cylindrical stem is 2 cm high with diameter 1.4 cm. Considering mushroom a solid object, answer the following questions :

(i) What is the total height of a mushroom ?

Hint:

Remember that for any hemisphere, the height from the flat base to the top of the dome is exactly equal to its radius!

Answer 1

Correct Answer: 5

Step 1: Identify the dimensions of the mushroom.
The mushroom consists of: (i) a hemispherical cap with radius \(R = 3\) cm, and (ii) a cylindrical stem with diameter \(= 1.4\) cm (so radius \(r = 0.7\) cm) and height \(h = 2\) cm.
Step 2: Find the total height of the mushroom.
Total height = height of hemisphere + height of cylinder \(= R + h = 3 + 2 = 5\) cm.
Step 3: Calculate the volume of the hemisphere.
\[V_{\text{hemisphere}} = \frac{2}{3}\pi R^3 = \frac{2}{3} \times \frac{22}{7} \times 27 = \frac{2 \times 22 \times 9}{7} = \frac{396}{7} \approx 56.57 \text{ cm}^3\]
Step 4: Calculate the volume of the cylinder.
\[V_{\text{cylinder}} = \pi r^2 h = \frac{22}{7} \times (0.7)^2 \times 2 = \frac{22}{7} \times 0.49 \times 2 = \frac{21.56}{7} \approx 3.08 \text{ cm}^3\]
Step 5: Calculate the total volume.
\(V_{\text{total}} = V_{\text{hemisphere}} + V_{\text{cylinder}} \approx 56.57 + 3.08 = 59.65 \text{ cm}^3\).
Step 6: State the answers.
\[ \boxed{\text{Total Height} = 5 \text{ cm}, \quad \text{Total Volume} \approx 59.65 \text{ cm}^3} \]